Denoted now with \(\Delta \doteq \mu _1 - \mu _0\)  our quantity of interest, its standard OLS estimator is given by \(\hat{\Delta }^{\text{OLS}} \doteq \hat{\mu }_1^{\text{OLS}} - \hat{\mu }_0^{\text{OLS}}\) . Notably, such estimator leads to invalid inference in the adaptive setting as demonstrated in [1]}. A valid alternative, with asymptotic theoretical guarantees is then proposed as follows.
